The Stacks project

Example 52.4.6. Let $k$ be a field. Let $A = k[x, y][[s, t]]/(xs - yt)$. Let $I = (s, t)$ and $\mathfrak a = (x, y, s, t)$. Let $X = \mathop{\mathrm{Spec}}(A) - V(\mathfrak a)$ and $\mathcal{F}_ n = \mathcal{O}_ X/I^ n\mathcal{O}_ X$. Observe that the rational function

\[ g = \frac{t}{x} = \frac{s}{y} \]

is regular in an open neighbourhood $V \subset X$ of $V(I\mathcal{O}_ X)$. Hence every power $g^ e$ determines a section $g^ e \in M = \mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{F}_ n)$. Observe that $g^ e \to 0$ as $e \to \infty $ in the limit topology on $M$ since $g^ e$ maps to zero in $\mathcal{F}_ e$. On the other hand, $g^ e \not\in IM$ for any $e$ as the reader can see by computing $H^0(U, \mathcal{F}_ n)$; computation omitted. Observe that $\text{cd}(A, I) = 2$. Thus the result of Lemma 52.4.5 is sharp.

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